Problem 4 - Essential Calculus

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{0}^{8} \sqrt[3]{x} d x, \quad n=8 $$

Step-by-Step Solution

Answer

After performing all the calculations according to the given rules and the exact definite integral, you should have three numerical values. By comparing these, you see how the Trapezoidal Rule and Simpson's Rule perform at approximating the integral in this case.

Step 1: Calculation using the Trapezoidal rule

First, we compute the approximation using the Trapezoidal Rule. Given the interval [a,b], function f(x), and n partitions, the Trapezoidal Rule follows the formula: \[ T_n = \frac{b - a}{2n} [ f(a) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(b)] \]Substituting the given values, we compute: \[ T_8 = \frac{8 - 0}{2*8} [ f(0) + 2f(1) + 2f(2) + ... + 2f(7) + f(8)] \]Plugging \[f(x) = \sqrt[3]{x}\] into the function arguments, perform the calculation and round the answer to four decimal places.

Step 2: Calculation using Simpson's rule

Now, compute the approximation using Simpson's Rule. Given the same inputs, the Simpson's Rule follows the formula: \[ S_n = \frac{b - a}{3n} [ f(a) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-2}) +2f(x_{n-1}) + f(b) ] \]Then, implement this formula with the given function and interval: \[ S_8 = \frac{8 - 0}{3*8} [ f(0) + 4f(1) + 2f(2) + ... + 4f(7) + f(8)] \]Substitute \[f(x) = \sqrt[3]{x}\] for the function arguments, calculate and round the result to four decimal places.

Step 3: Calculation of the exact definite integral

To find the exact value of the integral, we integrate the given function on the interval from 0 to 8. The integral of \[\sqrt[3]{x}\] is \[\frac{3}{4} x^{4/3}\], and using the Fundamental Theorem of Calculus, the definite integral \[ \int_{0}^{8} \sqrt[3]{x} dx \] becomes \[\frac{3}{4} * 8^{4/3} - \frac{3}{4} * 0^{4/3}\]. Perform this calculation and round the result to four decimal places.

Step 4: Comparison of the results

Finally, look at the results from the Trapezoidal Rule, Simpson's Rule, and the exact value of the definite integral. Observe how close or far the estimates from the rules are to the exact value. The smaller the difference, the more accurate the rule.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. In simple terms, it involves dividing the area under a curve into a series of trapezoids rather than rectangles. This approach can provide a more accurate approximation compared to basic methods like the Riemann sum.

Regarding the formula used for the Trapezoidal Rule, it is expressed as:

• \( T_n = \frac{b - a}{2n} [ f(a) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(b)] \)

Here, \(a\) and \(b\) represent the limits of integration, \(n\) is the number of subintervals, and \(f(x)\) is the function being integrated. This formula combines the function values at each subinterval, adjusting with a coefficient of 2 for interior points, giving you the approximate area.

When applying the Trapezoidal Rule to the given integral \( \int_{0}^{8} \sqrt[3]{x} \, dx \) with \(n = 8\), we plug in the function \(f(x) = \sqrt[3]{x}\) into our formula, essentially calculating the sum of areas of eight trapezoids. Don't forget to evaluate and round your answer to four decimal places for consistency.

Simpson's Rule

Simpson's Rule, another technique for numerical integration, takes a slightly more sophisticated approach. It considers not just straight-line segments like the Trapezoidal Rule, but parabolic arcs, offering potentially more accuracy.

The formula for Simpson's Rule is defined as:

• \( S_n = \frac{b - a}{3n} [ f(a) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-2}) + 2f(x_{n-1}) + f(b) ] \)

This works by applying alternate coefficients, 4 and 2, to the function values at the points, producing a weighted average that more faithfully represents the area under the curve.

For our example, Simpson’s Rule—using \( \int_{0}^{8} \sqrt[3]{x} \, dx\) and \(n = 8\)—requires us to compute function values of \(\sqrt[3]{x}\) at each subinterval point. We substitute these values into the formula, ensuring to multiply as specified. This can yield a result that is generally closer to the actual integral compared to the Trapezoidal approximation, especially important when accuracy is key.

Definite Integral

Finally, understanding the concept of a definite integral is fundamental when dealing with numerical integration methods. A definite integral represents the exact area under a curve within specified limits, \([a, b]\). In mathematical notation, a definite integral is expressed as:

• \( \int_{a}^{b} f(x) \, dx \)

The process involves finding an antiderivative of the function, followed by evaluating how much net space exists under the curve from the lower bound \(a\) to the upper bound \(b\).

In our exercise, the function \(f(x) = \sqrt[3]{x}\) is integrated over the interval [0, 8]. By applying the Fundamental Theorem of Calculus, we find:

• \( F(x) = \frac{3}{4} x^{4/3} \)
• \( \int_{0}^{8} \sqrt[3]{x} \, dx = \frac{3}{4} \times 8^{4/3} - \frac{3}{4} \times 0^{4/3} \)

This calculation gives us the exact value of the integral. Comparing exact values and approximations from the Trapezoidal and Simpson's Rule will help validate these numerical methods, highlighting the differences and accuracy levels.